Divide using synthetic division $$ ( x^{4}-5x^{3}+3x-4 ) \div ( x+3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&-5&0&3&-4\\& & -3& 24& -72& \color{black}{207} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{24}&\color{blue}{-69}&\color{orangered}{203} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-5x^{3}+3x-4 }{ x+3 } = \color{blue}{x^{3}-8x^{2}+24x-69} ~+~ \dfrac{ \color{red}{ 203 } }{ x+3 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&0&3&-4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&-5&0&3&-4\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&0&3&-4\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ -5 }&0&3&-4\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&0&3&-4\\& & -3& \color{blue}{24} & & \\ \hline &1&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}-3&1&-5&\color{orangered}{ 0 }&3&-4\\& & -3& \color{orangered}{24} & & \\ \hline &1&-8&\color{orangered}{24}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 24 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&0&3&-4\\& & -3& 24& \color{blue}{-72} & \\ \hline &1&-8&\color{blue}{24}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -69 } $
$$ \begin{array}{c|rrrrr}-3&1&-5&0&\color{orangered}{ 3 }&-4\\& & -3& 24& \color{orangered}{-72} & \\ \hline &1&-8&24&\color{orangered}{-69}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -69 \right) } = \color{blue}{ 207 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&0&3&-4\\& & -3& 24& -72& \color{blue}{207} \\ \hline &1&-8&24&\color{blue}{-69}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 207 } = \color{orangered}{ 203 } $
$$ \begin{array}{c|rrrrr}-3&1&-5&0&3&\color{orangered}{ -4 }\\& & -3& 24& -72& \color{orangered}{207} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{24}&\color{blue}{-69}&\color{orangered}{203} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-8x^{2}+24x-69 } $ with a remainder of $ \color{red}{ 203 } $.